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原子量の影響を考慮した液体 Sn 中における不純物拡散係数の予測式
Predicted Equation with Effect of Atomic Weight for
Impurity Diffusion Coefficients in Liquid Sn
椎木政人 1, 登林兼丸 1, 西村友希 1, 山中亜里紗 1, 鈴木進補 1
Masato SHIINOKI1, Kanemaru NOBORIBAYASHI1, Yuki NISHIMURA1, Arisa YAMANAKA1, andShinsuke SUZUKI1
1

早稲田大学，Waseda University

1. Introduction
Impurity diffusion coefficients in liquid metals are essential thermophysical properties for understanding
solidification. Our research group proposed a prediction method1) of impurity diffusion coefficient as deviatedvalues from the self-diffusion coefficient of the solvent based on the reliable data under the microgravity
condition and on the ground. However, it is necessary to include the effect of atomic weight in the predictedequation1) to improve the predictive accuracy. The objective of this study is to establish predicting methods
empirically for impurity diffusion coefficients including the effect of atomic weight in liquid Sn. The impurity
diffusion coefficients of Al, Au, and Cu in liquid Sn at 573 K were measured with high accuracy using theshear cell technique and stable density layering. Al and Au were selected as solute elements under theconditions that their atomic diameters are similar to that of Ag used in our previous experiment1), and their
atomic weights are smaller and larger than that of Ag, respectively. Cu was selected as the solute element
under the condition that its atomic diameter was sufficiently smaller than that of Sn.
2. Experimental procedure
A shear cell device2) with four capillaries was used. The detail design and procedure of the diffusion
experiments can be referred to previous expertiments1). The diffusion samples of Sn, Sn-Cu, Sn-Au, and Sn-Al
were prepared. The contents of the impurity elements were 1 and 3 at.% for Cu, 3 at.% for Au, and 2 at.% forAl. The diffusion temperature was 573 K and diffusion times tdiff were 28800 s for Cu and Au and 21600 s for
Al. When the samples were set into the shear cell device, the higher density samples were placed on the lower
side to obtain stable density layering.2) That is, Sn-Cu and Sn-Au alloys were set on the lower side, whereas
Sn-Al alloys were placed on the upper side. The diffusion axis was set parallel to the direction of gravity, andthe zero position of the x-axis was defined at the edge of the alloy sample. After homogenization, the diffusion
process started. At the end of the diffusion time, each capillary sample was mechanically separated into 20 cellsamples and was cooled down. The concentrations of Cu, Au, and Al in the cell samples were analyzed using
inductively coupled plasma optical emission spectroscopy (ICP-OES).
3. Results
Each concentration curve was obtained by fitting thick layer solution in Eq. (1) to 20 plots in each capillary.

1 of 3ci (x) =

c0
h0 +x
h0 -x
erf+erf
222√√〉〉{ ( 2〈Xmeas )( 2〈Xmeas )}(1)Here, ci is the concentration of impurity solute i, x is the position in the capillary, c0 is the initial concentrationof impurity solute i, h0 is the initial thickness of the alloy cell, and 〈X2meas 〉 is the measured mean-squarediffusion depth obtained from the diffusion experiments including systematic errors. In the fitting process, c0and 〈X2meas 〉 were varied as fitting parameters and h0 was set at a constant value of 3 mm as the thickness ofthe cell of the shear cell device. Diffusion coefficient was calculated by correcting the additional mass transport
as following equation;𝐷=〈X2meas 〉-〈X2add 〉2tdiff(2)

Here 〈X2add 〉 indicates the systematic error sources depended on the shear cell technique2). Figure 1 shows
the concentration profiles of Au at 573 K, which indicates that each obtained profile agrees with a thick layer
solution in Eq. (1). The other concentration profiles of Cu and Al also agreed with the thick layer solution inEq. (1). The averages of four measured impurity diffusion coefficients of Cu DCuSn, of Au DAuSn, and of Al DAlSn
were 2.88 × 10-9, 2.71 × 10-9, and 3.70 × 10-9 m2s-1, respectively.

4. Discussion
The concentration profiles of Au in Figure 1 showed that the experiment was highly reproducible. The
other concentration profiles of Cu and Au also showed these experiments were reproducible. In addition, therelative standard deviations in all experiments of Cu, Au and Al were much smaller than 6%: this value
includes the factors of uncertainty from temperature, duration, convection, and composition measurements,
in the microgravity experiment estimated by Garandet et al.3) The reproducibility of all experiments at 573 K
was higher than the estimated measurement error range of the microgravity experiment mentioned above. It
was therefore determined that all experiments at 573 K were performed with suppressed natural convection.
The values of DCuSn and DAuSn were close to the value of the self-diffusion coefficient of Sn4) D*Sn = 2.88 ×-9
10 m2s-1. Therefore, the condition for a discontinuity point in impurity diffusion coefficients in liquid Sn isthat (σSn/σi)ϕis is approximately 1.2 or larger. In the range of (σSn/σi)ϕis < 1.2, the effect of atomic weight iscorrected by the reduced mass ratio in predicted equation1): Predicted equation of impurity diffusion
coefficient Dis (s: solvent, i: impurity) is proposed to Dis = D*s(σs/σi)(M*ss /M*is )nϕis with the effect of atomic weight
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M*ss /M*is . Here, the self-diffusion coefficient D*s is the proportional constant, σi and ϕis are an atomic diameterand a thermodynamic factor, respectively. M*is (= (Mi Ms )/(Mi +Ms )) is a reduced mass, and Mi is an atomicweight. The obtained values of the exponent n were 0.364 by fitting the proposed equation to Dis measured in
Sn-Al and precious studies1) as following;
σM*

σiMis0.364

Dis = D*s ( s ) ( ss* )

ϕis .

(3)

Figure 2 shows the relationship between the proposed equation in n = 0.364 and the measured impurity
diffusion coefficients in liquid Sn at 573 K. The proposed equation was in good agreement with the Dis ofreference values with Al within 6 %.
5. Conclusions
The following different prediction methods for impurity diffusion coefficients of elements can be proposed
based on (σSn/σi)ϕis .
(1) When (σSn/σi)ϕis is approximately 1.2 or small:
The impurity diffusion coefficients in liquid Sn near the melting point can be estimated to be
proportional to the product of the following three factors with the self-diffusion coefficient of the Sn as
the slope: (i) the atomic diameter ratio of the Sn to the solute element σSn/σi, (ii) 0.364 power of thereduced mass ratio of the Sn to the solute element (M*SnSn /M*iSn )0.364, and (iii) thermodynamic factor ϕis .
(2) When (σSn/σi)ϕis is approximately 1.2 or large:
The impurity diffusion coefficients of elements in liquid Sn are close to the self-diffusion coefficient ofSn.
Acknowledgments
This work was supported by Grant-in-Aid for Scientific Research (C) Grant Number JP19K04990 andJP23K04364, Grant-in-Aid for Research Activity Start-up Grant Number JP24K23035, Grant-in-Aid for JSPS
Fellows Grant Number JP20J14950, Grant-in-Aid from the Mitsubishi Materials Corporation in the 2019 fiscal
year, German Aerospace Center and the German Academic Exchange Service (DLR/DAAD) ResearchFellowships (No. 57622550), and conducted as a part of the research project in 2019 for Research Assistant (Dr.
Masato Shiinoki) in Kagami Memorial Research Institute for Materials Science and Technology, Waseda
University.
References
1)

N. Yamada, S. Suzuki, K. Suzuki, A. Tanaka, R. Morita, C. Chenglin and G. Frohberg: Investigation of the Influenceof Different Solute on Impurity Diffusion in Liquid Sn using the Shear Cell Technique. Int. J. Microgravity Sci. Appl.,
35 (2018) 350402, DOI:https://doi.org/10.15011/jasma.35.4.350402.
2) S. Suzuki, K.-H. Kraatz and G. Frohberg: Ground-based diffusion experiments on liquid Sn-In systems using theshear cell technique of the satellite mission Foton-M1. Ann. N. Y. Acad. Sci., 1027 (2004) 169,
DOI:https://doi.org/10.1196/annals.1324.016.
3) J. P. Garandet, G. Mathiak, V. Botton, P. Lehmann and A. Griesche: Reference Microgravity Measurements of Liquid
Phase Solute Diffusivities in Tin- and Aluminum-Based Alloys. Int. J. Thermophys., 25 (2004) 249,
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© 2023 by the authors. Submitted for possible open access publication under theterms and conditions of the Creative Commons Attribution (CC BY) license(http://creativecommons.org/li censes/by/4.0/).

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